exponent where is the Lyapunov exponent and D is the diffusion coefficient. This scale measures the balance between stretching and diffusion on the evolution of the concentration field: stretching tends to decrease the width of a filament, while diffusion tends to increase it. The Batchelor scale is the smallest length scale that can be observed in the concentration field, since diffusion smears out quickly any finer detail. When most dye filaments reach the Batchelor scale, diffusion begins to decrease significantly the contrast of concentration between the filament and the surrounding domain. The time at which a filament reaches the Batchelor scale is therefore called its mixing time. The resolution of the advection–diffusion equation shows that after the mixing time of a filament, the decrease of the concentration fluctuation due to diffusion is exponential, resulting in fast homogenization with the surrounding fluid. History of chaotic advection The birth of the theory of chaotic advection is usually traced back to a 1984 paper by Hassan Aref. In this work, Aref studied the mixing induced by two vortices switched alternately on and off inside an inviscid fluid. This seminal work had been made possible by earlier developments in the fields of dynamical Systems and fluid mechanics in the previous decades. Vladimir Arnold and Michel Hénon had already noticed that the trajectories advected by area-preserving three-dimensional flows could be chaotic. However, the practical interest of chaotic advection for fluid mixing applications remained unnoticed until the work of Aref in the 80's. Since then, the whole toolkit of dynamical systems and chaos theory has been used to characterize fluid mixing by chaotic advection. Recent work has for example employed topological methods to characterize the stretching of fluid particles. Other recent directions of research concern the study of chaotic advection in complex flows, such as granular flows. References External links ctraj: Tools for studying chaotic advection. Chaos theory Fluid dynamics Turbulence